The campaigns of Donald Trump and Hillary Clinton acknowledged the huge stakes of Monday’s televised presidential debate, saying the event could shape the opinions of a large number of Americans who have yet to settle on their choice in what has been a bruising presidential election.

After more than a year of campaigning and attacking each other from afar, the two candidates will meet face to face for the first time, on a stage in Hempstead, N.Y. And as hard as it is to imagine, there remain enough undecided and loosely committed...